I'm aware of the Russian dominance of chess, but want to know just how this dominance came about. Thanks :)
I'll preface my answer by saying that I'm not a historian, but I'm a chess coach by trade so I feel qualified to answer this question.
Put simply, and as has been stated already, chess is simply taken much more seriously in the former Soviet countries than it ever has been in the west. Players who showed potential, regardless of country of origin within the USSR, got intense training at an early age. For example, the wiki on Boris Spassky mentions in passing that at age 10 he was "practicing for hours per day with master level coaches", as this was de rigeur for top level youth players in the USSR. There were thriving chess communities at the state universities in Leningrad and Moscow as well, and it wasn't unusual for promising players (Karpov for example) to study at one of these schools to be near their coach, and to be able to regularly compete against ranked opponents in chess tournaments.
There were a large variety of tournaments and opportunities for competition in the USSR, as well. Besides the standard USSR Chess Championship, there were citywide tournaments in Moscow, Leningrad and Kiev, in addition to a Women's national championship that ran from 1927 to 1991.
Along with the large degree of organization in competition, there was a developed and robust training and support community. Besides the support that would be shown to promising young players, mentoring and support was provided to Grandmasters throughout their career. I'm having a hard time finding a scholarly reference, but the Soviet Chess Federation headquarters in Moscow were legendary in the analog era for their library of games, commentary, and journals.
Finally, there was a level of mutual respect and friendly competition between the Grandmasters in the USSR. Whereas someone like Bobby Fischer was peerless in the US as a chess player, someone like Karpov or Kasparov has played countless top level players by the time they ever even play in a Candidates tournament, which is just for the privilege of challenging the world champion. When you practice every day against the best players in the world, it's a whole lot easier to become one yourself.
When you say Russian chess players do you mean players from the Russian SSR or from the Soviet Union as a whole? For example, Tal was Latvian, Keres Estonian, Petrosian Armenian and Kasparov, perhaps the greatest chess player of all time, was born in Azerbaijan to an Armenian mother and a Russian Jewish father.
Also, with the Soviet era I assume you mean post WW2? The Soviet chess wonder didn't start gaining strength until the 1930's and the first Soviet world champion was Botvinnik in 1948 (not counting Alekhine who became a French citizen the same year he won the world championship).
Here's an old post on this question. /u/mijumarublue's reply is particularly focused on the Russian side of things.